1. **State the problem:** We are given two functions $f(x) = 4x + 3$ and $g(x) = x + 5$. We need to find the product function $(f \cdot g)(x)$, which means multiplying $f(x)$ and $g(x)$. 2. **Formula used:** The product of two functions $f$ and $g$ is defined as: $$ (f \cdot g)(x) = f(x) \times g(x) $$ 3. **Calculate the product:** Substitute the given functions: $$ (f \cdot g)(x) = (4x + 3)(x + 5) $$ 4. **Expand the product:** Use distributive property (FOIL method): $$ (4x + 3)(x + 5) = 4x \times x + 4x \times 5 + 3 \times x + 3 \times 5 $$ $$ = 4x^2 + 20x + 3x + 15 $$ 5. **Combine like terms:** $$ 4x^2 + (20x + 3x) + 15 = 4x^2 + 23x + 15 $$ 6. **Restrictions on the variable:** Since $f(x)$ and $g(x)$ are polynomials, there are no restrictions on $x$. The domain is all real numbers. **Final answer:** $$(f \cdot g)(x) = 4x^2 + 23x + 15$$